Are we technically able to make, in hardware, arbitrarily large neural networks with current technology?
The approach you describe is called neuromorphic computing and it's quite a busy field. 

IBM's TrueNorth even has spiking neurons. 

The main problem with these projects is that nobody quite knows what to do with them yet. 

These projects don't try to create chips that are optimised to run a neural network. That would certainly be possible, but the expensive part is the training not the running of neural networks. And for the training you need huge matrix multiplications, something GPUs are very good at already. (Google's TPU would be a chip optimised to run NNs.)

To do research on algorithms that might be implemented in the brain (we hardly know anything about that) you need flexibility, something these chips don't have. Also, the engineering challenge likely lies in providing a lot of synapses, just compare the average number of synapses per neuron of TrueNorth, 256, and the brain, 10,000.

So, you could create a chip designed after some neural architecture and it would be faster, more efficient, etc …, but to do that you'll need to know which architecture works first. We know that deep learning works, so google uses custom made hardware to run their applications and I could certainly imagine custom made deep learning hardware coming to a smartphone near you in the future. To create a neuromorphic chip for strong AI you'd need to develop strong AI first.